The Egyptians were familiar with remainders from division, and this naturally led them to the concept of fractions. The Egyptians decided to only use unit fractions (that is, fractions with numerator 1) to represent numbers, although there is evidence that 2/3 and 3/4 were permitted.
Non-unit fractions were represented as the sum of distinct unit fractions.
For example, 2/7 was written as 1/4 + 1/28. Note that 1/7 + 1/7 is not the sum of distinct unit fractions.
We notice that the expressions are not unique. For example: 7/24 = 1/6 + 1/8 = 1/4 + 1/24.
A problem arising from this system is working out how to write any fraction as a sum of unit fractions. It is unclear how the Egyptians resolved this problem, but J. J. Sylvester (1814 – 1897) presented the following algorithm for finding the unit fraction expansion of a number between zero and one:
1. Find the largest unit fraction less thn the given fraction.
2. Subtract this fraction from the given fraction, and repeat.